Invariant measure

Definition

Let (X, \Sigma) be a measurable space and let f : X \to X be a measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be invariant under f if, for every measurable set A in \Sigma, \mu\left(f^{-1}(A)\right) = \mu(A).

In terms of the pushforward measure, this states that f_*(\mu) = \mu.

The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted M_f(X). The collection of ergodic measures, E_f(X), is a subset of M_f(X). Moreover, any convex combination of two invariant measures is also invariant, so M_f(X) is a convex set; E_f(X) consists precisely of the extreme points of M_f(X).

In the case of a dynamical system (X, T, \varphi), where (X, \Sigma) is a measurable space as before, T is a monoid and \varphi : T \times X \to X is the flow map, a measure \mu on (X, \Sigma) is said to be an invariant measure if it is an invariant measure for each map \varphi_t : X \to X. Explicitly, \mu is invariant if and only if \mu\left(\varphi_{t}^{-1}(A)\right) = \mu(A) \qquad \text{ for all } t \in T, A \in \Sigma.

Put another way, \mu is an invariant measure for a sequence of random variables \left(Z_t\right)_{t \geq 0} (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition Z_0 is distributed according to \mu, so is Z_t for any later time t.

When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1, this being the largest eigenvalue as given by the Frobenius–Perron theorem.

Examples

• Consider the real line \R with its usual Borel σ-algebra; fix a \in \R and consider the translation map T_a : \R \to \R given by: T_a(x) = x + a. Then one-dimensional Lebesgue measure \lambda is an invariant measure for T_a.
• More generally, on n-dimensional Euclidean space \R^n with its usual Borel σ-algebra, n-dimensional Lebesgue measure \lambda^n is an invariant measure for any isometry of Euclidean space, that is, a map T : \R^n \to \R^n that can be written as T(x) = A x + b for some n \times n orthogonal matrix A \in O(n) and a vector b \in \R^n.
• The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points \mathbf{S} = {A,B} and the identity map T = \operatorname{Id} which leaves each point fixed. Then any probability measure \mu : \mathbf{S} \to \R is invariant. Note that \mathbf{S} trivially has a decomposition into T-invariant components {A} and {B}.
• Area measure in the Euclidean plane is invariant under the special linear group \operatorname{SL}(2, \R) of the 2 \times 2 real matrices of determinant 1.
• Every locally compact group has a Haar measure that is invariant under the group action (translation).

References

• John von Neumann (1999) Invariant measures, American Mathematical Society

References

1. {{wikibooks-inline. Geometry/Unified Angles